We study the homology of Riemannian manifolds of finite volume that are covered by a product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of the real hyperbolic plane. Using a variation of a method developed by Avramidi and Nyguen-Phan, we show that any such manifold $M$ possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic $r$-dimensional submanifolds whose fundamental classes are linearly independent in the real homology group $H_r(M;\mathbb{R})$.

中文翻译：

---

双曲平面乘积覆盖的黎曼流形中同调类的几何构造

我们研究由乘积 $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ 覆盖的有限体积黎曼流形的同调性实双曲平面。使用由 Avramidi 和 Nyguen-Phan 开发的方法的变体，我们证明任何这样的流形 $M$ 拥有任意大量的紧凑定向平面完全测地线 $r$ 维子流形，其基本类是在实同调群 $H_r(M;\mathbb{R})$ 中线性无关。